Polymer 58 (2015) 170e179
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Polymer
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Visualization of micro-scale inhomogeneities in acrylic thickener
solutions: A multiple particle tracking study
A. Kowalczyk, C. Oelschlaeger*, N. Willenbacher
Karlsruhe Institute of Technology (KIT), Institute of Mechanical Process Engineering and Mechanics, Gotthard-Franz-Str.3, 76128 Karlsruhe, Germany
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 1 September 2014
Received in revised form
17 November 2014
Accepted 20 December 2014
Available online 27 December 2014
Multiple Particle Tracking (MPT) microrheology in combination with bulk mechanical rheometry has
been used to characterize the structural and viscoelastic microheterogeneity of commercial acrylic
thickeners Carbopol EDT2050, Viscalex HV30, and Sterocoll D in aqueous solutions at technically relevant
polymer concentrations. According to bulk rheology Carbopol solutions exhibit a much higher elasticity
than Sterocoll and Viscalex solutions at low polymer concentration whereas all thickeners are composed
of similar main monomers. MPT experiments confirm that the latter two systems form homogeneous
predominately elastic polymer networks with a mesh size <200 nm. In contrast, Carbopol solutions are
highly heterogeneous and the degree of heterogeneity strongly increases with increasing polymer
concentration. Predominantly elastic and viscous regions within the solution can be identified based on
the slope of the mean squared displacement of individual particle trajectories. This heterogeneity is
directly imaged here for the first time using Voronoi diagrams and characteristic length scales varying
from 5 to 20 mm are found. Additionally, variation of the probe size reveals that the elastic regions
themselves are heterogeneous including areas with mesh size <200 nm and a larger fraction with mesh
size between 200 nm and 500 nm.
© 2014 Elsevier Ltd. All rights reserved.
Keywords:
Microrheology
Multiple particle tracking (MPT)
Heterogeneous fluids
1. Introduction
Synthetic acrylic polymers are frequently used as thickening
agents in water-based coatings and adhesives or personal care
products. Typically, these commercial alkali-swellable acrylates
[1e4] (as well as various other polymeric thickeners) form inhomogeneous partly aggregated or cross-linked solutions. Inter- and/
or intramolecular aggregation is due to hydrophobic groups
randomly distributed along the chains and the swelling behavior
can be varied via solvent quality [5e7]. Crosslinking can be induced
either thermally or by adding appropriate crosslinking agents during synthesis. Accordingly, such thickener solutions cover a wide
range of rheological behavior, ranging from weakly elastic, almost
Newtonian to highly elastic gel-like. Despite its high technical
relevance, little is known so far about the contribution of the microscale inhomogeneity to the bulk viscoelastic properties [8]. Here we
use the method of Multiple Particle Tracking (MPT) to quantify the
degree of structural and mechanical microheterogeneity of such
* Corresponding author.
E-mail address: claude.oelschlaeger@kit.edu (C. Oelschlaeger).
http://dx.doi.org/10.1016/j.polymer.2014.12.041
0032-3861/© 2014 Elsevier Ltd. All rights reserved.
acrylic thickener solutions. This technique was originally described
by Apgar et al. [9] and Ma et al. [10]. Up to now MPT has been
frequently used to study microheterogeneities of actin filament
network [11e14], living cells [15e19], proteins [20,21], DNA solutions [22], biological gels [23,24] or yield stress fluids [1e3]. The
fluid mechanics of microrheology has been described thoroughly
[25,26]. The basic idea is to monitor the Brownian motion of inert
fluorescent tracer particles by means of digital video microscopy. For
homogeneous fluids the bulk shear moduli [27] can be determined
using this technique, but more importantly the statistical analysis of
tracer trajectories allows for a characterization of sample inhomogeneity. Al least some hundred particles have to be monitored
simultaneously to allow for a significant statistical analysis in order
to cover a sufficiently large area of the samples. For a better interpretation and characterization of the heterogeneous dynamics and/
or microstructure, statistical distributions of displacement and local
viscosity [11,22,23,28,29] but also van Hove correlation plots and the
non-Gaussian parameter a [30e33] are used to characterize the
heterogeneous and microstructure of fluids. The non-Gaussian part
of the van Hove correlation function Gs(x,t) [32,33] is defined as the
probability density distribution of the displacements of individual
particles [32,34]:
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
Gs ðx; tÞ ¼
1 XN
Nðx; tÞ
⟨ i¼1 d½x þ xi ð0Þ xi ðtÞ⟩ ¼
N
N
(1)
with X the distance of particle center of mass along the X -coordinate. N(X,t) is the number of particles found at positions between
X and (X þ d X) after lag time t and N is the total number of particles. If all particles are exposed to similar environment Gs(x,t) has
a Gaussian form. Deviations from this form reflect the presence of
heterogeneities. Such deviations can be characterized by the nonGaussian parameter
a¼
⟨x4 t ⟩
3x2 ðtÞ2
1
(2)
P
b
with ⟨xb ðtÞ⟩ ¼ N
i¼1 xi ðtÞ*Gs ðx; tÞ the simplest combination of the
second (b ¼ 2) and fourth (b ¼ 4) moments of a one-dimensional
probability density function. a is zero for a Gaussian distribution,
while distributions result in large values of a [30,33].
In the first part of the paper, we characterize bulk rheological
properties of the three different acrylic thickeners and investigate
variations of yield stress sy and plateau modulus G0 as a function of
polymer concentration. In the second part, we describe and use the
MPT method to perform microrheological measurements on
thickener solutions presenting similar bulk elastic properties, i.e.
similar G0 values. For those samples, we generate the variation of
mean square displacements (MSD) as a function of time and from
statistical analysis of the MSD distribution we extract microstructural and micro-mechanical heterogeneity informations.
Additionally, for Carbopol only we investigate probe size and
polymer concentration effects on microrheological properties in
order to get additional microstructural information. Finally, we use
Voronoi diagrams constructed using the particle positions as generators in order to visualize the size and spatial distribution of
heterogeneities based on a “rheological contrast”, i.e. classification
of particle mobility and hence viscoelasticity of its surrounding
according to the slope of the respective MSD.
2. Materials and methods
2.1. Samples
Our experiments were performed using three acrylic thickeners,
Sterocoll D, Viscalex HV 30 (both from BASF SE Ludwigshafen,
Germany) and Carbopol ETD 2050 (Lubrizol Advanced Materials,
Cleveland, USA). For brevity, we use the family name of each
product in the subsequent text. These are commercial, alkaliswellable thickeners based on homopolymers and co-polymers of
polyacrylic acid and a small amount of a crosslinking agent [7,35].
Upon neutralization the weak acrylic acid groups dissociate in
aqueous environment, the polymer chains get soluble and the
thickening properties are developed. Sterocoll is mainly used as
rheology modifier for rotogravure and paper coating [36] and Viscalex for water-borne coating [37,38]. Both are synthesized in an
emulsion polymerization process and delivered as a milky liquid
with a solids content of 25%e30% and pH z 2.5e3. In both cases,
main monomers are ethylacrylate (EA) and methacrylic acid (MAA).
Sterocoll is an alkali swellable emulsion polymer (ASE) with a MAA/
EA molar ratio of about 50:50 and a small fraction of diethylenically
unsaturated monomer as chemical crosslinker. Viscalex differs in a
sense that it is a hydrophobically modified alkali swellable emulsion (HASE). It contains hydrophobic alkyl groups attached to its
polymer backbone. The molar ratio MAA/EA/Alkyl is about 49:50:1.
In aqueous solution, hydrophobic association junctions are formed
providing side chains physical crosslinking. Structure representations for these types of polymers are shown in studies of Ng et al.
171
[6], Dai et al. [39] and Wang et al. [5]. For the investigations presented here, aqueous solutions of Sterocoll with concentrations of
0.25e5wt. % and Viscalex with concentrations of 0.25e1.5 wt.%
were prepared. Solutions were stirred at room temperature for 48 h
and adjusted to pH ¼ 8 by slowly adding 1 N NaOH [4,7]. Carbopol
ETD 2050 consists of high molecular weight polymers, made up of
homopolymers of acrylic acid and copolymers of acrylic acid and
long chain (C10eC30) alkyl acrylate crosslinked with a polyalkenyl
polyether. Illustration of the Carbopol structure is shown in
Ref. [40]. The carboxylic groups are the principle chemical sites that
affect the thickening characteristics. It is supplied as powder. We
prepared samples (0.1e1wt.%) by slowly adding powder in water,
which decreases the pH to 3 while stirring, consequently carboxylic
groups release the hydrogen atom. The mixture was further stirred
at room temperature for 48 h and adjusted to pH ¼ 6 by slowly
adding 4 N NaOH. Neutralization replaces a free hydrogen cation
with the sodium cation and the repulsive force between the negatives charges of the carboxyl groups and osmotic pressure from
mobile ions cause the structure to swell, resulting in a highly elastic
microgel system optically transparent, but known to exhibit an
inhomogeneous structure on the micrometer scale [1,2]. Increasing
the pH further does not change the viscoelastic elastic properties
very much. Carbopol is used in applications including clear gels,
hydroalcoholic gels and lotions [41].
2.2. Mechanical measurements
A rotational rheometer Thermo MARS II equipped with a coneplate measuring cell (diameter dCP ¼ 60 mm, cone angle acone ¼ 1 )
was used to perform steady as well as small amplitude oscillatory
shear experiments. The latter covering the frequency range from
0.01 to 100 rad.s1. Strain sweep experiments performed prior to
frequency sweeps ensure that the strain amplitude used was sufficiently small to provide a linear material response at all investigated frequencies. The yield stress was determined by fitting a
HerscheleBulkley model to the shear rate/shear stress data obtained from stress ramp experiments covering the stress interval
from 0.1 Pa to 50 Pa within a measuring time of 1800 s. All measurements were performed at 20 C and a solvent trap was used to
avoid evaporation of the solvent during the experiments.
2.3. Multiple-particle tracking (MPT) setup
MPT experiments were performed using an inverted fluorescence microscope (Axio Observer D1, Zeiss), equipped with a Fluar
100, N.A. 1.3, oil-immersion lens combined with a 1x optovar
magnification changer. We have tracked the Brownian motion of
green fluorescent polystyrene microspheres of 0.19, 0.51 mm
diameter (Bangs Laboratories: USA, lot Nr FC03F/7049) used as
tracer particles. The mixture (total volume: ~20 ml) containing the
sample solution including the tracers (volume fraction around 1%)
was injected into a commercial rectangle capillary (CM Scientific
Ltd, United Kingdom). The sample thickness was ~150 mm and the
microscope was focused roughly halfway into the sample to minimize wall effects. Images of the fluorescent beads were recorded
onto a personal computer via a sCMOS camera Zyla X (Andor
Technology: 21.8 mm diagonal sCMOS Sensor size, 2560 2160
square pixels (6.5 mm), up to 50 frames/s in global shutter mode).
Displacements of particle centers were monitored in an 84 84 mm
field of view, for at least 100 s at a rate of 30 frames/s. For each
experiment a total of roughly 300 particles were tracked simultaneously. Movies of the fluctuating microspheres were analyzed by a
custom MPT routine incorporated into the software Image Processing System (Visiometrics iPS) and a self-written Matlab program [42]. Tracing errors due to particles moving in and out of the
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A. Kowalczyk et al. / Polymer 58 (2015) 170e179
focal plane are a fundamental problem of two dimensional MPT
microrheology; these errors have been eliminated using an
advanced tracing algorithm called radius of gyration (RG) criterion
which is based on the widely used Crocker and Grier tracking algorithm [43]. The particle displacements between subsequent images are identified as so-called artificial jumps if this displacement
deviates more than four standard deviations from the mean value
and trajectories are terminated at such positions. In the subsequent
processing step trajectories which are separated by a time gap not
larger than Dtmax are merged based on an adaptive search radius
criterion taking into account a reduced radius of gyration and the
individual particle mobility. Data analysis relies on a reasonable
choice of Dtmax and the accessible time scale depends on a minimum number of time steps or length Lmin of the trajectories
accepted for further analysis. This data processing and analysis
scheme is thoroughly described in Ref. [42]. Additionally, a homogenous distribution of the tracer particles embedded in the
probe is necessary to allow for a meaningful interpretation of the
microstructure of the investigated thickener solutions. This has
been evaluated calculating the function g(r) from all particle positions within an image. Corresponding results are shown in Fig. 1 for
solutions of all three thickeners types investigated here. In all cases
an almost ideal behavior with g(r) z1 is found, indicating a homogenous particle distribution in all samples.
3. Results and discussion
3.1. Bulk rheology measurements
Characterization of bulk viscoelastic properties of the three
acrylic thickener solutions has already been performed in many
rheological studies [1e3,7,42,44e46]. These solutions exhibit a
strong gel like behavior and an apparent yield stress. Even at low
polymer concentration of 1% we found yield stress values of
sy ¼ 10.3 ± 0.6, 6.0 ± 0.6 and 2.2 ± 0.5 Pa for Carbopol, Viscalex and
Sterocoll, respectively. Fig. 2A shows the plateau modulus G0 ¼ G0
(u ¼ 1 rad.s1) determined from oscillatory shear measurements as
a function of polymer concentration for the three thickeners.
An important observation is that at low polymer concentration
G0 is much higher for Carbopol (squares) than for Viscalex
(triangles) and Sterocoll D (circles) solutions, e.g. a concentration of
0.25% G0 ~ 25 Pa for Carbopol whereas it is only 1 and 0.05 Pa for
Viscalex and Sterocoll, respectively. This strong difference is
intriguing especially from a technical point of view as the three
thickeners comprise with similar main monomers. Another difference resides in the concentration dependence of G0. For Carbopol
this variation is described by a simple power-law dependance
G0 ~ c3/4 linear over the full concentration range investigated. The
moduli of Viscalex and Sterocoll exhibit a much stronger concentration dependence especially at concentration below 1% but start
at much lower values. In order to gain a better understanding of
these bulk rheological differences, we have investigated the
microrheological properties of three thickener solutions with
similar bulk elastic properties; i.e. similar plateau modulus G0,
using MPT. Solutions of 0.3% Carbopol, 0.5% Viscalex and 2% Sterocoll exhibit G0 values between 21 and 29 Pa considered to be equal
within experimental error (see hatched box in Fig. 2A). As can be
seen from Fig. 2B the variation of dynamic shear moduli G0 and G00
with frequency is almost similar in the whole frequency range
investigated (0.01<u < 100 rad.s1). The storage modulus G0 is always larger than the loss modulus G00 (G0 G00 ) and no crossover is
observed indicating a very long terminal relaxation time (>100 s).
Nevertheless, we observe a weak G0 increase as a function of u for
Sterocoll and Viscalex solutions. The latter result may be explained
by higher molecular weight polydispersity compared to Carbopol.
Chains of different length whose stress contributions will relax at
different rates lead to this slight G0 dependence. Additionally, by
extrapolating G0 and G00 at low frequencies, we can predict a G0 , G00
crossover that will occur at a certain frequency uc for both Sterocoll
and Viscalex solutions indicating a terminal flow regime. Terminal
shear relaxation times for Sterocoll and Viscalex solutions have
been reported in Martinie et al. [47] and Kowalczyk et al. [48]. For
both samples, at the polymer concentrations investigated here,
relaxation times of about TR ¼ 1/uc ~ 900 s were found. For Carbopol solutions, no relaxation time data are found in the literature
but according to Fig. 2B, TR must be much larger than for the
Sterocoll or Viscalex solutions investigated here if it exists at all. It
seems that the heterogeneous structure with its sample-spanning
network of highly elastic regions hinders or at least delays stress
relaxation compared to the homogeneous Viscalex and Sterocoll
solutions. Finally, we can directly determine the mesh size x of the
polymer network from the plateau modulus value G0. According to
the classical theory of rubber elasticity G0 ¼ kBT/x3 [49] and
G0 ¼ 25 Pa corresponds to x ¼ 55 nm. This value is much smaller
than the diameter of the embedded tracer particles (~500 nm) used
for MPT measurements. Therefore, we can treat the medium as a
continuum around the embedded probe particles.
3.2. Microrheology measurements
Fig. 1. Pair correlation function g(r) as a function of distance r for Viscalex 0.5% (dashed
line), Sterocoll 2% (dotted line) and Carbopol 0.3% (full line) solutions.
As mentioned previously, microrheological measurements have
been performed on samples showing similar bulk elastic properties. Fluorescent polystyrene microspheres ~500 nm in diameter
have been used as tracer particles. Particle trajectories were constructed using the track algorithm in combination with the RG
criterion described in Section 2C. The corresponding MSD traces for
Carbopol, Sterocoll and Viscalex are shown in Fig. 3A-C,
respectively.
For Carbopol (Fig. 3A), the range of displacement at a given lag
time is very broad, e.g. t ¼ 1/10 s, MSD varies more than two orders
of magnitude, from 7.105 to 8.103 mm2. The lowest MSD value of
7.105 mm2 obtained at the shortest lag time t ¼ 1/30 s exceeds the
resolution of our setup which is about 3.105 mm2 thus the so called
static error correction as described in Ref. [50] can be neglected. A
substantial fraction of MSD traces adopt a power-law behavior as a
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
173
Fig. 2. A) Plateau modulus G0 as a function of acrylic thickener concentration. Hatched box highlights the three samples investigated with MPT. B) Frequency dependence of shear
moduli G0 (closed symbols) and G00 (open symbols) for Sterocoll 2% (circles), Viscalex 0.5% (triangles) and Carbopol 0.3% (squares).
function of time ⟨Dr2(t)⟩ftb with a slope b close to 1 throughout
the probed time scales. This result indicates that the motion of
these beads is purely diffusive and that the microenvironment
surrounding the particles responds like a viscous liquid. Other MSD
traces exhibit a curvature and the power law exponent tends towards b ¼ 0 at long lag times indicating that the particles are highly
constrained by the surrounding solution. In that case, the motion is
subdiffusive which is the signature of elastic trapping of the beads
in the solution. This result clearly reveals a strong heterogeneity of
the Carbopol solution on the mm length scale as already reported by
Oppong et al. [2,3] who suggest that the microstructure of Carbopol
gels consists of concentrated centers of strongly cross-linked
polymer molecules surrounded by regions of more dilute polymer. Roberts et al. [44] proposed the structure of an existing highlycrosslinked core with overlapping dangling ends in a viscous
environment. Another structure has been suggested by Kim et al.
[46] based on cryo-scanning electron microscopy (cryo-SEM).
These images suggest a transition from a string-like entangled
structure to a honeycomb-like rigid structure (pore size ~5 mm)
with increasing polymer concentration.
In Sterocoll solution (Fig. 3B) microspheres move significantly
different. Here almost all MSDs depend only weakly on time (b
~0.1) for t < 10s, indicating that particles are highly constrained by
the surrounding solution. This result is consistent with an elastic
trapping of tracer particles in a homogeneous gel-like environment.
At long times (t > 10 s) the slope of the MSD traces approaches
b ¼ 1 indicating slow viscous diffusion of the beads corresponding
to a transition into the terminal flow regime. Compared to the
Carbopol solution, the range of displacements at a given time is
much narrower; it varies e.g. from 3.104 to 2.103 mm2 at t ¼ 0.1 s.
Similar behavior is observed for the Viscalex solution (Fig. 3C). In
this case MSD traces exhibit a plateau throughout all the probed
time scales, no MSD increase is observed at long lag times indicating a longer relaxation time tR > 30s. Moreover, similar MSD
variations have been obtained for Sterocoll and Viscalex solutions
using tracer particles with diameter 200 nm. This result indicates
that the mesh size of the polymer network in both systems is
clearly below 200 nm which is in good agreement with the mesh
size value xbulk ¼ 55 nm determined from bulk G0 data based on
classical rubber elasticity theory. In summary, MPT measurements
show that Carbopol is strongly inhomogeneous on the mm-scale
whereas Sterocoll and Viscalex are both homogeneous solutions at
the polymer concentrations investigated here.
3.2.1. Van Hove correlation function
In Fig. 4 we report the van Hove correlation functions in order to
perform the statistical analysis of the distribution of displacements
for Carbopol 0.3%, Sterocoll 2% and Viscalex 0.5% solutions calculated at a lag time t ¼ 3s. This lag time value has been chosen
arbitrarily as the non-Gaussian parameter a extracted from this
analysis is almost independent of time (result not shown). For the
Viscalex solution (blue cross) we observe that the MSD data are
Fig. 3. Mean square displacement of microspheres (~500 nm) dispersed in aqueous solution of Carbopol 0.3% (A), Sterocoll 2% (B) and Viscalex 0.5% (C). The white curve is the
ensemble-average MSD.
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A. Kowalczyk et al. / Polymer 58 (2015) 170e179
Fig. 4. Van Hove correlation functions for the ensemble particles at time lag t ¼ 3s for
Carbopol (black), Sterocoll (red), Viscalex (blue). The lines show the Gaussian fit to the
data.
described by a Gaussian distribution obtained at a given lag time
indicating that all tracer particles exhibit similar diffusivity. The low
value of a ¼ 0.15 is a strong indicator of a homogeneous system. For
Sterocoll (red cross) we find a ¼ 0.4. Although this value is slightly
higher than that obtained for the Viscalex solution, it still confirms
a high degree of homogeneity of the system. Finally, for the Carbopol solution (black cross) we observe a strong deviation of the
measured displacements from a Gaussian distribution with a ¼ 4.
This again clearly manifests the high degree of heterogeneity
compared to the other two solutions.
Table 1 shows the variation of a for all three solutions as a
function of polymer concentration.
For Sterocoll and Viscalex solutions a is independent of polymer
concentration within experimental error. In both cases a values are
less than one (~0.5) indicating homogeneous systems independent
of polymer concentration. On the contrary, for the Carbopol solution a increases from ~1.5 to ~7 when the polymer concentration
increases from 0.1 to 0.75%, respectively. This strong increase in a
indicates that the Carbopol micro-structure becomes more and
more heterogeneous as the polymer concentration increases. This
will be discussed in more detail in the next section.
3.2.2. MSD distributions and statistical analysis
To quantify the level of inhomogeneity in the different solutions
we generate the MSD slope dMSD/dt distributions (Fig. 5) and MSD
distributions (Fig. 6) calculated from the individual microspheres
trajectories at a lag time t ¼ 3s. For both Sterocoll (Fig. 5B) and
Table 1
Non-Gaussian parameter a for all three thickeners in solutions with different
polymer concentration.
Carbopol
wt%
a
Sterocoll
wt%
a
Viscalex
wt%
a
0.1%
0.3%
0.75%
1.6 ± 0.26
3.5 ± 0.65
6.8 ± 0.86
0.25%
0.5%
2%
0.2 ± 0.06
0.35 ± 0.13
0.4 ± 0.08
0.5%
1%
1.5%
0.15 ± 0.09
0.15 ± 0.10
0.39 ± 0.09
Viscalex (Fig. 5C) solutions, the MSD slope distribution is narrow
with almost all values <0.5 and <0.3, respectively indicating a homogeneous strongly elastic environment. On the contrary, for the
Carbopol solution (Fig. 5A) the distribution of the dMSD/da values
is very broad and asymmetric with approximately 70% of MSD data
having a slope <0.5, indicating a more elastic environment and
30% > 0.5 corresponding a more viscous surrounding.
This result is confirmed by the distribution of absolute MSD
values obtained at a lag time t ¼ 3s shown in Fig. 6AeC.
For the Carbopol solution (Fig. 6A) the MSD distribution is
bimodal with a majority of particles having MSD absolute values
around 102 mm2 and a second population with a broad distribution
of MSDs centered around 101 mm2. For Sterocoll (Fig. 6B) and
Viscalex (Fig. 6C) solutions, the distribution is narrow and in both
cases all MSD absolute values are well below 102 mm2.
Fig. 7AeC characterize the micro-structure of the solutions by
plotting all particles positions in the plane of observation; color
coding each individual position according to the MSD slope
assigning a circular area for each data point corresponding to the
MSD of the respective particle. This representation allows us to
physically map the variation in the local mechanical response
across the sample and to directly get information about the distribution of elastic and viscous regions throughout the sample. It
should be noted that this representation is different from that
introduced by Valentine et al. [13] where in their study trajectories
are color coded according to cluster analysis. They define a cluster
as a group of particles where all the particles in a given cluster are
statistically indistinguishable. In a first approach, we clearly see
that the displacement of embedded beads in a Carbopol solution
(Fig. 7A) is completely different from that observed in Sterocoll
(Fig. 7B) and Viscalex (Fig. 7C) solutions. For the two latter solutions, almost all particles show a small dot size and a similar color
code (red). This representation again confirms that both samples
are highly homogeneous elastic materials on the length scale of the
probe particles (~500 nm). On the contrary, the map indicates a
strong heterogeneity of the Carbopol solution in terms of color code
and dot size characterizing the mobility of the individual particles.
In this case, the sample is inhomogeneous on the mm length scale
with particles that move freely within viscous region of about
10 mm in size surrounded by elastic areas where tracer mobility is
highly restricted.
3.2.3. Variation of polymer concentration
In this section, we discuss the effect of polymer concentration
on microrheological properties in order to obtain additional information about the structure. Fig. 8 shows the variation of the scaling
exponent b of the ensemble-average MSD ~ tb as a function of
Sterocoll and Viscalex concentration.
For Sterocoll, b decreases continuously from about one to zero
when increasing the polymer concentration from 0.25 to 3%. At the
same time, the values for the non-Gaussian parameter a are found
to be less than one independent of Sterocoll concentration within
experimental error (see Table 1). This result indicates a transition
from a homogeneous Newtonian fluid to a homogeneous highly
elastic gel-like material. A uniform network forms with gradually
decreasing mesh-size as polymer concentration increases. Similar
results have been obtained for Viscalex solutions, i.e. the nonGaussian parameter (a~0.2) is independent of Viscalex concentration, but b decreases monotonically with polymer concentration,
again indicating a uniform homogeneous network structure. In
both systems, the tracer particles used (diameter 500 nm) are much
larger than the mesh size of the polymer network. A different
completely behavior is obtained for the Carbopol solutions when
varying the polymer concentration. At 0.1% concentration (Fig. 9A),
the MSD slope distribution shows a majority of particles (~85%)
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
175
Fig. 5. Distributions of MSD slopes measured at time lag t ¼ 3s 3s for microspheres (0.5 mm) dispersed in an aqueous solution of Carbopol 0.3% (A), Sterocoll 2% (B) and Viscalex 0.5%
(C).
Fig. 6. MSD distributions measured at time lag of t ¼ 3s for microspheres (0.5 mm) dispersed in an aqueous solution of Carbopol 0.3% (A), Sterocoll 2% (B) and Viscalex 0.5% (C).
with slopes >0.5 and a peak around 0.75, indicating that almost all
particles see a predominantely viscous environment. At a concentration of 0.3% (Fig. 9B), the solution becomes heterogeneous with
~65% of particles being in an elastic environment (dMSD/da <0.5)
and ~35% moving freely (see comments of Fig. 5A). Finally at a
concentration of 0.75% (Fig. 9C), only ~15% of particles move freely
and a majority (~85%) is confined in a strongly elastic environment.
Strong heterogeneities of such Carbopol solutions and a similar
variation of dMSD/da distributions as a function of polymer concentration have been observed by Oppong et al. [3] in their MPT
experiments with tracer particles of diameter 1 mm. In their study
no probe size variation has been performed. Furthermore small
angle light scattering experiments [51] have shown that Carbopol
solutions consist of microgel particles with a strongly crosslinked
core approximately 500 nm in diameter surrounded by a highly
expanded shell of polymer chains ~20 mm in size. This shell has a
low monomer concentration as well as a low crosslink and/or
entanglement density. The highly crosslinked core of the microgel
particles is not accessible for the tracer particles but they can move
diffusively through the shell. Accordingly, at low Carbopol
Fig. 7. Visualization of heterogeneity: x, y positions of individual embedded beads, with corresponding mean free path (dot size) multiplied by a factor of 40, and corresponding
slope of MSD (color scale) for Carbopol (A), Sterocoll (B) and Viscalex (C).
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A. Kowalczyk et al. / Polymer 58 (2015) 170e179
35% for tracers with diameter 520 nm. Corresponding nonGaussian parameter values are ~0.5 and ~4, respectively. The fact
that the majority of particles with diameter 190 nm moves freely
indicates that the mesh size in most of the elastic, entangled regions is greater than 200 nm. This further elucidates the heterogeneity of the solution consisting of few regions with a mesh size
smaller than 200 nm and a major fraction of areas with mesh size x
between 200 and 500 nm as schematically sketched in Fig. 11. This
is in fair agreement with the average mesh size xMPT ¼ (180 ± 50)
nm calculated from the MSD traces of the 510 nm particles with
slope close to zero (see Table 2 next session).
Fig. 8. Variation of the scaling exponent b of the ensemble-average MSD as a function
of Sterocoll (circles) and Viscalex (squares) concentrations. Diameter of the tracer
particles was 500 nm.
concentrations the solution appears homogeneous, the a-parameter is low and essentially all tracer particles explore a viscous
environment (see Fig. 10). As the polymer concentration increases,
the microgel particles start to interpenetrate thus forming elastic
regions with high entanglement or crosslinks density. These elastic
regions grow and the fraction of tracer particles in a viscous environment decreases with increasing polymer concentration.
Accordingly, the non-Gaussian parameter characterizing solution
heterogeneity increases with increasing polymer concentration as
shown in Fig. 10. A schematic sketch of this change in microstructural heterogeneity is provided in Fig. 11. Finally, the fraction of
particles in a viscous environment reaches a minimum value of
about 0.2 at a polymer concentration of 0.75% and the nonGaussian parameter levels off at a z 7.
Further information about the complex microstructure of Carbopol solutions is gained from additional MPT measurements using
tracer particle of the same nature but with a lower diameter of
190 nm. Preliminary measurements ensured that these smaller
tracer particles were homogeneously distributed by calculating the
pair correlation function g(r) which was almost equal to 1 independent of r. We focused on the sample with Carbopol concentration 0.3% and found that approximately ~80% of tracer particles
with diameter 190 nm are in a viscous environment compared to
3.2.4. Heterogeneity length scale of Carbopol solutions
A direct visualization of the solution microstructure can be obtained by a Delaunay triangulation of the focal plane images using
the average positions of the tracer particles as generators. Around
each particle a so-called Voronoi cell is constructed including all
points closer to that particle than to any other particle in the plane
[52]. Fig. 12AeE shows Voronoi diagrams for Carbopol 0.1, 0.2, 0.3,
0.4 and 0.75% solutions, respectively. These representations allow
for a direct mapping of the solutions microstructure in terms of
viscous (white) and elastic (black) regions corresponding to dMSD/
dt > 0.5 and dMSD/dt < 0.5, respectively. In other words, polymer
concentration fluctuations in the solutions are imaged here for the
first time based on a “rheological contrast”. At low Carbopol concentration (0.1%), a majority of the solution is viscous with very few
elastic inclusions. As the polymer concentration increases the total
fraction of viscous areas decreases continuously, finally reaching a
structure consisting of isolated individual viscous areas of about
5e10 mm in diameter separated from each other by ~10e20 mm at
0.75% polymer concentration. Fig. 12F shows the fraction of image
area attributed to viscous regions as a function of Carbopol concentration. This quantity decreases from 85% to 21% when Carbopol
concentration increases from 0.1% to 0.75% respectively. At the
same time, the number of individual, disconnected viscous regions
increases from 1 to 40.
Table 2 shows the variation of the apparent viscosity happ,
plateau modulus G0 and mesh size xMPT with Carbopol concentration as determined from MPT measurements. The quantity happ
corresponds to the apparent viscosity in the viscous regions and
has been determined from MSD traces with slope >0.5 using the
!2
relation ⟨D r ðtÞ⟩ ¼ 4Dt in combination with the StokeseEinstein
relation D ¼ kBT/6pha including the diffusion coefficient D and the
tracer particle radius a. This latter equation holds for particles freely
diffusing in an infinite medium. In our case viscous regions are
surrounded by highly elastic areas. These boundaries modify the
hydrodynamics of the diffusing particles and essentially lead to a
Fig. 9. Distribution of MSD slopes measured at time lag of 3s for microspheres (500 nm) dispersed in aqueous solutions of Carbopol 0.1% (A), 0.3% (B) and 0.75% (C).
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
177
Table 2
apparent viscosity happ, plateau modulus G0 and mesh size xMPT as determined from
MPT measurements for Carbopol solutions of different viscosity.
Fig. 10. Fraction of particles in a viscous environment (squares) and non-gaussian
parameter a (triangles) as a function of Carbopol concentration.
decrease of the diffusion coefficient as it has been discussed in the
literature extensively [53e56]. Accordingly calculated viscosity is
higher than the true viscosity of the surrounding liquid and thus is
termed apparent viscosity happ in the following. The storage
modulus G0 has been determined from MSD traces with slope close
to zero, i.e. from particles moving in a predominantly elastic environment using the relationG0;app ¼
size is then obtained from
2kB T
!2
. The apparent mesh
3pa⟨D r ðtÞ⟩
G0;app ¼ xk3B T according
MPT
to the classical
rubber elasticity.
We observe a continuous increase of happ in the viscous regions
from 45 to 148 mPa as Carbopol concentration increases from 0.1 to
0.75%. This increase is supposed to be due to an increase of polymer
concentration within the viscous regions, but it has to be kept in
mind that these viscous regions shrink in size (see Fig. 12 AeE) and
hence the measured diffusion coefficient of the tracer particles may
decrease due to the above mentioned confinement effects. The
Brownian motion of particles in the vicinity of a wall has been
studied, both theoretically [53] and experimentally [54,55]. Lançon
et al. [56] have shown that the diffusion coefficient of spherical
colloidal particles between parallel walls strongly decreases when
the channel width approaches the particle diameter. They report a
30% reduction of the diffusion coefficient perpendicular to the wall
for a confinement ratio of 10, i.e. when the distance between the
walls is 10 times the particle diameter and a 50% reduction for a
Conc./wt%
happ/mPas
0.1
0.2
0.3
0.4
0.75
45
46
66
89
148
±
±
±
±
±
16
23
27
37
85
G0,app/Pa
0.27
0.59
1.06
1.85
8.5
±
±
±
±
±
0.2
0.5
0.8
1.6
2.0
xMPT/nm
281
231
179
160
78
±
±
±
±
±
101
82
49
59
36
confinement ratio of 5. This might serve as a reasonable estimate of
the confinement effect here and would correspond to an increase of
apparent viscosity by a factor of 1.33e2, which is not so far from
what is observed when Carbopol concentration is increased from
0.1 % to 0.75 % and may account at least partly for the observed
increase in happ.
The plateau modulus G0,app in the elastic regions increases from
0.27 to 8.5 Pa in the above mentioned concentration range indicating an increasing crosslink density. The corresponding mesh size
x decreases from 281 to 78 nm. These values are much higher than
those obtained from bulk measurements, i.e. for Carbopol 0.3%,
xBulk ¼ 55 nm < xMPT ¼ 179 nm, and this discrepancy is attributed to
the inhomogeneity of the material. The xMPT values are much
smaller than the diameter of the tracer particles and characterize
the elastic properties and network structure in the vicinity of the
corresponding tracer particles. Based on light scattering data [51]
and our MPT results especially the Voronoi diagrams we can
describe the microstructure of Carbopol solutions in different
concentration regimes. At low concentration individual microgel
particles consisting of a highly cross-linked core regions surrounded by an expended shell roughly 20 mm in diameter are
present in the solution. The cross-link or entanglement density in
this shell is so low that the tracer particles can diffuse freely, only a
small fraction of tracers is elastically trapped presumably in regions
where microgel particles interpenetrate and the mesh size if the
polymer network decreases well below 500 nm (see Fig. 12A).
When increasing the polymer concentration, overlap among
microgel particles increases and corresponding elastic regions form
an interconnected sample-spanning network. At 0.2% polymer
concentration both the elastic regions as well as the viscous regions
are connected among themselves throughout the sample (see
Fig. 12B). But at higher polymer concentration the viscous regions
are isolated and shrink more and more in size as shown in
Fig. 12CeE.
4. Conclusions
MPT in combination with classical bulk mechanical rheometry
has been used to characterize the structural and viscoelastic
microheterogeneity of three commercial acrylic thickeners in
Fig. 11. Illustrations representing the microstructure of Carbopol at concentrations of 0.1% (A) and 0.3% (B) with tracer particles of diameter 510 nm, 0.3% with tracer particles of
diameter 190 nm (C). Black dots represent the highly crosslinked cores of the microgels, the wiggly lines the surrounding shell. Blue dots represent tracer particles in a viscous
environment and red ones particles trapped in elastic regions. The number of particles corresponds to the relative values given in Fig. 10.
178
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
Fig. 12. Voronoi diagrams for Carbopol 0.1% (A), 0.2% (B), 0.3% (C), 0.4% (D) and 0.75% (E) based on experiments with 510 nm tracer particles. Viscous areas are in white and elastic
areas in black. Number of individual viscous region (square) and viscous region in % (triangle) as a function of Carbopol concentration (F).
aqueous solutions at technically relevant polymer concentrations.
From bulk rheology we found for Carbopol solutions a higher
elasticity at low polymer concentration and a weaker dependence
of the storage modulus on concentration than for the Sterocoll and
Viscalex solutions. MPT experiments confirm that these two latter
systems form pre-dominantly elastic polymeric network with a
mesh size <200 nm in the concentration range investigated here. In
contrast, Carbopol solutions are highly heterogeneous on a micrometer length scale and the degree of heterogeneity as characterized
by the non-Gaussian parameter a obtained from a van-Hove analysis of MPT data strongly increases with increasing polymer concentration. Predominately elastic and viscous regions within the
solution can be identified based on the slope of the mean squared
displacement (MSD) of the individual particle trajectories. This
heterogeneity is directly imaged here for the first time using Voronoi diagrams. Carbopol solutions consist of microgel particles with
a highly crosslinked core (500 nm in diameter) and a dilute shell
(20 mm in diameter) with a mesh size larger than the tracer particles used in our MPT experiments. Elastic regions are formed when
microgel particles overlap to reduce the mesh size of the polymer
network below the size of the probe particles (500 nm). At low
Carbopol concentration the solutions are predominately viscous
with few elastic inclusions typically 10 mm in size. At intermediate
concentrations sample-spanning, connected elastic and the viscous
regions interpenetrate. Finally, at high polymer concentration
viscous inclusions are found within an elastic matrix and the size of
these inclusions decreases with increasing polymer concentration
with a characteristic length scale of about 500 nm at 0.75% Carbopol concentration. Variation of the probe size reveals that the
elastic regions themselves are heterogeneous including areas with
mesh size <200 nm and a larger fraction with mesh size between
200 nm and 500 nm. The bulk modulus G* of such heterogeneous
systems depends on the specific properties of the predominantly
elastic and viscous regions as well as the characteristic length scale
of the corresponding concentration fluctuations. Development of
an appropriate model for calculation of the bulk modulus G*(u)
based on the micro-structural and micro-rheological data obtained
from MPT experiments will be targeted in future research. This is of
high technical relevance since the bulk modulus is tightly related to
the thickening and application properties of such polymers and a
correlation between microstructure, microrheology and bulk
rheology is the key to a rational goal-oriented product design.
Acknowledgments
The authors thank BASF-SE (Ludwigshafen, Germany) and
Lubrizol Advanced Materials (Cleveland, USA) for providing the
three acrylic thickeners.
Appendix A. Supplementary data
Supplementary data related to this article can be found at http://
dx.doi.org/10.1016/j.polymer.2014.12.041.
A. Kowalczyk et al. / Polymer 58 (2015) 170e179
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